Quantum inspired approach for denoising with application to medical imaging

Background noise in many fields such as medical imaging poses significant challenges for accurate diagnosis, prompting the development of denoising algorithms. Traditional methodologies, however, often struggle to address the complexities of noisy environments in high dimensional imaging systems. This paper introduces a novel quantum-inspired approach for image denoising, drawing upon principles of quantum and condensed matter physics. Our approach views medical images as amorphous structures akin to those found in condensed matter physics and we propose an algorithm that incorporates the concept of mode resolved localization directly into the denoising process. Notably, our approach eliminates the need for hyperparameter tuning. The proposed method is a standalone algorithm with minimal manual intervention, demonstrating its potential to use quantum-based techniques in classical signal denoising. Through numerical validation, we showcase the effectiveness of our approach in addressing noise-related challenges in imaging and especially medical imaging, underscoring its relevance for possible quantum computing applications.


IntroducAon
Imaging noise, such as speckle noise in ultrasound, Gaussian noise in magneYc resonance imaging (MRI), and sway of electronic noise, sca(er radiaYon, random coincidences, and a(enuaYon effects in nuclear imaging can obscure important anatomical structures and details, making it difficult for clinicians to accurately interpret images and make informed diagnoses.By effecYvely removing this noise, denoising algorithms enhance the clarity and fidelity of medical images, improving their diagnosYc uYlity and enabling clinicians to idenYfy subtle abnormaliYes or pathologies more accurately.In recent years, many computaYonal techniques such as neural networks [1][2][3][4] , regularizaYon-based techniques [5][6][7] , and staYsYcal approach 8,9 have shown posiYve progress on addressing the challenges of image denoising and background noise reducYon.However, despite the success of these methodologies, they exhibit deficiencies when confronted with the complexiYes inherent in noisy image environments.For instance, neural networks when dealing with learning intricate pa(erns and representaYons, ocen struggle with the detailed structures present in noisy images, especially in domains like medical imaging where data acquisiYon is constrained in imaging modaliYes due to design or target applicaYons such as limited angle tomography [10][11][12][13] or Compton camera [14][15][16] .Furthermore, the reliance on large annotated datasets for training can be prohibiYve, hindering the robustness and generalizaYon capabiliYes of neural networks, parYcularly in the presence of significant noise levels.AddiYonally, neural networks suffer from overfidng, parYcularly when trained on small datasets or noisy images, leading to subopYmal denoising performance.Similarly, while regularizaYon-based approaches provide a framework for controlling computaYonal model and prevenYng overfidng, they may struggle to capture the diverse and complex structures present in noisy images due to non-linearity and inability to capture local extrema, or complex regularizaYons, thereby limiYng their effecYveness in noise reducYon tasks.These limitaYons underscore the need for alternaYve methodologies that can effecYvely address the complexiYes of denoising in high dimensional medical imaging while ensuring robustness and generalizaYon across diverse imaging modaliYes and noise condiYons.
In recent years, a few a(empts have been made to apply quantum principles in image or signal processing, including early work 17 and proceeding efforts in image segmentaYon 18,19 .More recent developments [20][21][22][23] adopt a quantum inspired approach to imaging systems.These methods show promising start for the uYlizaYon of quantum physics into the denoising problems.One important aspect of those works was to process images as block-wise to preserve pixel correlaYon for efficient denoising, unlike previous methods that begin with a conYnuous mathemaYcal representaYon and then discreYze.Also, it was shown that one can use the quantum localizaYon phenomenon and quantum interference effects to address noise dependence; however previous works do not fully exploit localizaYon in terms of modal analysis.AddiYonally, like neural network and regularizaYon approaches, prior quantum-inspired methods rely on a set of hyperparameters regarding key parameters in both quantum mechanics theory and the filtering process, rendering them as quasi-automaYc approaches requiring some manual intervenYon.With the imminent rise of quantum computaYon, there is a growing need for standalone quantum approaches that effecYvely address these issues.
In this paper, we extend on the previous work 20 on quantum inspired approach for denoising problems.We introduce a contrasYng view of the medical image where we idealize a medical image as amorphous (disordered) structure akin to the condensed ma(er physics, and we use the amorphous model analysis to characterize the locality and propagaYng behaviors of the signals in decomposed imaging systems.In this view, we consider a noiseless image as a localized structure with the absence of diffusive behaviors, and therefore, in contrast, we consider the background noise as diffusive and non-local modal representaYves.We will show that in various examples, similar characterizaYon of the disordered regime is relevant in imaging space and is parYcularly useful for denoising.While the primary focus of this paper is on medical imaging, we also include examples of classical natural images to enable all readers to be(er evaluate its performance.Overall, the contribuYon of this paper lies in different points: -We present a contrasYng view where we draw a parallel between an image (e.g.medical image) and the theory of amorphous structures in condensed ma(er physics.-Thresholding and filtering are conducted in the quantum domain, using rules inspired by the laws of physics.
-The core of the proposed method is the Schrödinger's equaYon.Here, the Planck constant is defined within a physical context as opposed to exisYng methods.-The approach eliminates the need for hyperparameter opYmizaYon, presenYng a standalone quantum-inspired method.
-It achieves a reducYon in computaYonal costs.The paper proceeds with an organized examinaYon of our proposiYon, beginning with an exposiYon on the basics of quantum mechanics.It then delves into modal analysis techniques applied to amorphous structures in condensed ma(er, offering insights into the characterizaYon of vibraYonal modes.Next, the paper introduces quantum-inspired denoising methodologies, tailored specifically for image systems depicted as amorphous structures.Subsequently, it discusses the connecYon between the amorphous model and images affected by background noise, laying the groundwork for denoising in compressed sensing applicaYons and filtering process.Through various examples, the paper demonstrates the effecYveness of these techniques in enhancing image quality and compressing the essenYal image components.

Basics of Quantum Mechanics
Quantum physics theory explores the behavior of parYcles at the quantum scales, challenging the classical understanding of the surrounding phenomena.This theory has been crucial for understanding the properYes of complex systems such as solids, liquids, and gases.At the heart of quantum physics lies Schrödinger's equaYon, a fundamental equaYon that describes how the wave funcYon of a quantum system evolves over Yme.In a non-relaYvisYc single parYcle quantum system, a wave funcYon () describes the probability of presence of a parYcle in a potenYal (), where  is the spaYal posiYon.This wave funcYon is an element of a Hilbert space with bounded integrals and follows the staYonary Schrödinger equaYon 24 : () = () (1) where  = (−ℏ 2 ⁄ )∇ !" + () is the Hamiltonian operator.Here  and  are the mass and the energy of the parYcle and ℏ is the Planck constant that relates the energy of the parYcle to its frequency, ∇ !" is spaYal Laplacian derivaYves at spaYal posiYons .
In condensed ma(er physics, the amorphous structure is ocen locally described by harmonic oscillators and the potenYal energy funcYon takes the form of () = # "  "  " where ω is the angular frequency of the oscillator.SubsYtuYng this potenYal into the Schrödinger equaYon yields the following: This equaYon is known as the Yme-independent Schrödinger equaYon for the harmonic oscillator.It describes the energy levels and wave funcYons of a quantum harmonic oscillator system.The soluYons to this equaYon give the quanYzed energy levels of the harmonic oscillator, which are equally spaced, and the corresponding wave funcYons represent the probability distribuYons of finding the parYcle at various posiYons along the oscillator.Thus, from the Schrödinger equaYon with the harmonic oscillator potenYal, we obtain a dynamical system of the quantum harmonic oscillator.The harmonic oscillator model can provide insights into the vibraYonal behavior of amorphous materials.In this context, the vibraYonal modes are not strictly phonons, but rather collecYve excitaYons involving the moYon of atoms or molecules within the material.These excitaYons can sYll be approximated as harmonic oscillators, with each mode characterized by a specific frequency and associated energy.
In imaging examples, we assume that the pixels are the parYcles, and the potenYal is described by the intensity values of the pixels (i.e., for 2D image,  = , where ∈ ℝ $×$ represents the intensity value of image containing  ×  pixels).Therefore, the Schrödinger equaYon yields the following form: and similarly, this results in a dynamical system where the soluYon is the set of eigenvectors that serve as an adapYve basis in decomposed images or signals.As discussed in previous works, for a vectorized 2D image, when the convenYonal zero padding is used as boundary condiYons for the Hamiltonian operator, the discreYzed Hamiltonian matrix,  ∈ ℝ $ !×$ !, takes the simplified form of where (, ) is the (, )-th component of the Hamiltonian operator.Unlike in quantum mechanics, the Planck constant in image processing is a parameter that should be tuned.Previous studies have opYmized the value of the Planck constant manually to choose the opYmal one in order to denoise an image.Here we propose a formula to esYmate the opYmal value based on quantum mechanics: where  and  are the energy and frequency of the decomposed image.Here the denominator represents the maximum frequency of an image denoted by a half of pixel number in one direction, /2, and the numerator is denoted by the total normalized squared intensity of an image.Furthermore, we define ℏ 6 = ℏ where  ∈ ℕ as an augmented value to study its influence on our problem.

Amorphous Regime in Disordered Harmonic Solids
In amorphous regime, materials exhibit a lack of long-range order in their atomic structure, sedng them apart from crystalline solids.Within this context, the atomisYc vibraYonal eigenmodes in amorphous materials can be classified into two main categories: propagaYng and non-propagaYng modes 25 .PropagaYng modes, characterized by longer wavelengths, are wavelike vibraYonal movements through a homogeneous medium and they are undiscerning of atomisYc disorder.On the other hand, non-propagaYng modes are called diffusons and locons.Diffusons extend across the enYre amorphous sample, represenYng vibraYonal eigenmodes that diffuse energy over the material without being confined to specific regions.In contrast, locons are spaYally localized modes, where vibraYonal movements are trapped within specific regions of the material.In essence, the vibraYonal modes of atoms in amorphous materials can be broadly categorized into three types: propagons, which propagate and are non-local; diffusons, which do not propagate but are non-local; and locons, which are localized and non-propagaYng [25][26][27] .Figure 1 demonstrates the behaviors of these three vibraYonal modes in amorphous structure.  of the disorder.For disYncYon between propagons and diffusons, a well-known Ioffe-Regel criterion 28 comes into play.
According to this criterion, as the mean free path of a parYcle, such as a phonon, approaches the magnitude of the interatomic spacing or the size of the material's structural units, a transiYon occurs in the vibraYonal modes from propagons, indicaYve of propagaYng states, to diffusons, represenYng diffusive or non-propagaYng states.For diffusons-locons separaYon, Anderson localizaYon 29 provides a useful picture which refers to the phenomenon where wavefuncYons become localized in a disordered medium, prevenYng the propagaYon of waves over long distances.In the case of vibraYonal modes in amorphous materials, Anderson localizaYon can lead to the trapping of vibraYonal energy within specific regions due to the disorder in the atomic arrangement.The disYnguishing criterion for separaYng diffusons from locons is commonly referred to as the mobility edge 25 .
In terms of computaYonal tools, the parYcipaYon raYo serves as one of the indicators of localized modes and Anderson localizaYon phenomena in amorphous materials.A low parYcipaYon raYo means that an eigenstate is highly localized, indicaYng that its amplitude is concentrated within a small number of elements.Conversely, a high parYcipaYon raYo implies that the eigenstate is more spread out or delocalized, with its amplitude distributed across numerous elements.For vibraYonal modes, the parYcipaYon raYo is given as follows: ) & (6)   where  ⃗ <,> is the eigenvector for mode  at atom  and  is the total number of atoms.
To demonstrate the relevance of the parYcipaYon raYo and criteria for disYnguishing vibraYonal modes in amorphous structures, we computed parYcipaYon raYo for a common amorphous Silicon (a-Si) structure that inhibits the short-range order (SRO), which means a length scale smaller than 5 Å, while lacking long-range order 30 .We employed a previously developed conYnuous random network (CRN) 31 as an illustraYve example.The atomisYc structure generated from the CRN uYlizes a random-based atomic arrangement method with a bond-swapping algorithm.The CRN framework constructs the structure with SRO and preserves disorder beyond the second neighbor lengths, resulYng in the eliminaYon of defects and voids.Specifically, the CRN structure of a-Si exhibits a defect and void concentraYon of less than 1-3%.The simulated system contains 4096 atoms and the Tersoff potenYal 32 is used in GULP package 33 to obtain eigenvectors by solving the dynamical system.Figure 2 shows the parYcipaYon raYo for example a-Si structure.The two shaded regions indicate the separaYon of propagons at low frequencies from diffusons and diffusons from locons at higher frequencies.While diffusons dominate the frequency range the propagons and locons count for small fracYon of the total modes.We should note that the separaYon criteria are subjecYve and usually mulYple factors are considered to obtain a clear-cut separaYon.However, the calculaYon for parYcipaYon raYo is straighporward with relaYvely low computaYonal cost and this behavior is rather consistent between all amorphous structures.

Amorphous Regime and LocalizaEon in Image Processing
Images are broadly considered as spaYally structured data with localized pa(erns.This prompted us to explore whether mode resolved localizaYon characterizaYon tools are applicable for addressing medical imaging problems and other scenarios where spaYal localizaYon within the image is a key aspect.Here we invesYgate the presence of localizaYon in image with noise and noiseless structures using the concept of mode resolved parYcipaYon raYo, inspired with the analogy with amorphous solids.Similar to condensed ma(er analysis of vibraYonal modes, we calculate the parYcipaYon raYo of the sample image as following: ) # (7)   where  ⃗ .,> is the eigenvector for mode  and pixel number  and  is total number of pixels.
The top row of Figure 3 shows the benchmark syntheYc image of the size 64×64 pixels (which results in the same total number of modes as a-Si in Figure 2.) and noisy images with Poisson noise at signal to noise raYo (SNR) of 2, 5, and 15.The eigenvectors and eigenvalues of the syntheYc image and noisy ones were calculated from dynamical system, equaYons 3 and 4, at esYmated Planck constant of 0.7864 and subsequently the normalized parYcipaYon raYo, the normalizaYon is defined as the parYcipaYon raYo over the total number of modes (pixels).The bo(om plots in Figure 3 show the parYcipaYon raYos for the corresponding syntheYc image and noisy ones.InteresYngly the pa(ern of localizaYon and parYcipaYon raYo resemble the similar behavior as vibraYonal modes in amorphous regime.The comparison between parYcipaYon raYo results shows that as SNR increases, the degree of the localizaYon is diminished, and the mode resolved parYcipaYon raYo uniformly increases parYcularly for the mid-range eigenvalues.In parallel, the idenYficaYon of localized modes becomes more and more clear for low and high eigenvalues as SNR increases.This implies that the increase in noise level increases the parYcipaYon of the non-local modes and ulYmately results in a more clear-cut separaYon of the low and high localized modes.For large noise, e.g.SNR=2, there is a clear localizaYon of the low and high eigenvalues, and the mid values dominate the eigenvalue domain.To further demonstrate the similariYes between the vibraYonal modes in amorphous regime and mode resolved image, we show the plots of selected eigenvectors through three cross-secYons [corresponding to the first to third rows of the image] for low, mid, and high eigenvalues in Figure 4. Three cases show the propagaYng and non-local behavior for the low range modes, sca(ered and non-localized behavior for mid-range modes and, localized with non-propagaYng nature for high-range modes.Therefore, with comparison with Figure 1, the eigenvectors of imaging system are comparable to amorphous structures in condensed ma(er physics.We should note that although the same characterisYcs are apparent between the mode resolved image regime and vibraYonal modes in amorphous regime, the similar naming of propagon, diffuson and locons may not be appropriate in studying imaging systems.Also, we emphasize that the quantum localizaYon premise in imaging systems relies on the spaYal localizaYon of the image's structural components.Hence, this characterisYc can be effecYvely uYlized for image processing.In the next secYon, we will discuss the connecYon between localizaYon and background noise, and we propose our approach of uYlizing localizaYon to remove noise and compress the image reconstrucYon.

ApplicaEons to Denoising and Compressed Sensing
Besides the environmental condiYons and the limitaYons on the measurement systems, background noise ocen refers to unwanted signals or interference that diffuses through the image due to the sca(ering of incident radiaYon or waves by parYcles or structures within the imaged area.Hence, we consider background noise as a non-localized mode that is parYcipant in the characterisYcs of many pixels, and they represent the sca(ered behavior similar to the diffusons in vibraYonal modes of amorphous structures.To examine this proposiYon, we use a modified version of the quantum inspired approach 20 for denoising process that was constructed by smoothing the input signal, compuYng the eigenmodes, manually thresholding them, and then back-projecYng the thresholder eigenmodes.In the modified approach, we introduce a criterion to remove/filter the non-localized modes with high parYcipaYon raYo based on the histogram distribuYon plot of the normalized parYcipaYon raYo.Given that the majority of modes are non-local modes with high parYcipaYon raYo, we fit a Lorentzian funcYon on the spectral distribuYon to select these non-local modes.The Lorentzian funcYon, Φ, as a funcYon of parYcipaYon raYo is defined as where the  F is the peak parYcipaYon raYo value, Γ is the half width at half maximum (HWHM).Figure 5 shows the distribuYon of the parYcipaYon raYo for the noisy syntheYc image of SNR=2 and fi(ed Lorentzian funcYon on the peak distribuYon in the outset.The corresponding region below the Lorentzian funcYon fit gives a good approximaYon of the high parYcipaYon and non-local modes.Here we consider the lower bound of the Lorentzian fit and the eigenvalues corresponding to the low and high values as the threshold to filter the signals for both low and high range modes.The inset of the figure 5 shows the selected region that is used in the modified approach.This results in fixing an effecYve value for the thresholding in the adapYve basis, which is in this way determined from the noisy image directly.The summary of presented modificaYons is summarized in Algorithm 1. Algorithm 1: denoising algorithm using the proposed approach Input: Image,  1. Compute the Planck constant using equation ( 5) and Hamiltonian matrices, , using equation ( 4) 2. Calculate the eigenvectors of equation ( 3) and singular value decomposition (SVD) 3. Calculate the participation ratio using equation ( 7) 4. Fit the Lorentzian fit using equation ( 8) to the distribution of normalized participation ratio and threshold based on the image modes with non-local and non-propagating modes 5. Recover the denoised image using the filtered image Output: Denoised image,  6 Figure 6 shows the denoised images using the full set of eigenvectors based on the previous work 20 and filtered denoised images are the results of modified quantum approach using only low and high range modes which are shaded in the corresponding parYcipaYon raYo of noisy image, these results are shown for three SNR's of 2, 5, 10, and 15.AddiYonally, histograms of the normalized parYcipaYon raYo are provided for further clarificaYon on its behavior.In all cases, we observe the comparable results between the modified approach, which include only shaded region, versus the approach with all mode inclusion.To clarify the differences Table I  The values of SSIM's are slightly smaller for lower SNR values while slightly larger for the higher SNR values.Overall, the comparison between two approaches, filtered and non-filtered, shows that the contribuYon of the mid-range eigenvectors is minimal in the reconstrucYon process and the majority of those modes are related to the added noise.Also, we observe that as the noise level increases, the efficacy of the filtering becomes more effecYve.

The Role of Planck Constant on LocalizaEon in Imaging System
The Planck constant is not a well-established concept for imaging systems.To understand the influence of the Planck constant on the localization, we calculated the participation ratio based on the augmented Planck constants, ℏ 6 = ℏ , as assumed for  =1, 2, and 4 [where for  = 4 the ℏ 6 is near to the previously used Planck constant].Figure 7 shows the corresponding results for the syntheYc image with SNR=2, for the augmented Planck constants.

Results for the complex image
Here, we present the results for a complex image (Lena image) with a size of 256×256 pixels.We compare the outcomes obtained through the original method, where all modes are utilized for the denoising process, with those achieved through the modified approach, which involves selecting specific modes for denoising and reconstruction, as shown in figure 8.The added noise follows a Poisson distribution with an SNR of 15.

Figure 8.
ComparaYve analysis of denoising methods: quantum inspired approach with all modes vs. modified approach with selected high localizaYon modes.The figure illustrates (a) the Lena image and its zoom-in counterpart (a'), (b) the noisy image and its zoom-in counterpart (b'), (c) the denoised image using all modes and its zoom-in counterpart (c'), and (d) the denoised image using the modified approach and its zoom-in counterpart (d').
The results illustrate that the denoising process yields comparable results for the modified version when compared to considering all modes.Specifically, the denoising results exhibit a slight decrease in contrast, accompanied by a slightly smoother appearance compared to the original denoising approach.The SSIM values for the original and modified approaches are 0.76 and 0.75, respecYvely, while the PSNR values for the original and modified approaches are 28.15 dB and 27.85 dB, respecYvely.
AddiYonally, Figure 9 shows the normalized parYcipaYon raYo for the noisy Lena image presented in Figure 8-(b).The shaded regions represent the corresponding low and high eigenvalue ranges based on the Lorentzian fit of the parYcipaYon raYo distribuYon following the modified approach.We observe a nearly symmetrical parYcipaYon raYo between low and high eigenvalues, with greater localizaYon and disYnct separaYon into the high parYcipaYon raYo (mid-eigenvalue) region.These results suggest that for complex images, localizaYon is notably more pronounced compared to simple structured images.Results for the medical image Lastly, we compare the results obtained between original and modified methods for a computed tomography (CT) image with a size of 260×260 pixels in Figure 10.
raYo vs. eigenvalues for a noisy CT image with SNR=5, and (c) normalized parYcipaYon raYo vs. eigenvalues for a noisy CT image with SNR=15; shaded regions show the low and high eigenvalues that are selected in modified approach.For SNR=5, the figure displays (d) the noisy CT image, (e) the denoised image using all modes, and (f) the denoised image using the modified approach.For SNR=15, it shows (g) the noisy CT image, (h) the denoised image using all modes, and (i) the denoised image using the modified approach.
The results show the success of the modified approach to capture the comparaYve results in all cases while about 70% of all mid-range eigenvalues are dismissed.To compare two approaches in terms of quality metrics, Table 2 summarizes the SSIM and PSNR values; those values consistently show the excellent performance for the modified approach and point to the success of the criteria to choose the effecYve modes based on the quantum localizaYon for the medical image example.AddiYonally, we tabulated the simulaYon Ymes of both approaches for the CT image in Table 3.Each simulaYon was conducted using in-house MATLAB code on a single node of Bridges2 (with the regular memory 256 GB) at the Pi(sburgh SupercompuYng Center, and a cluster with 2 AMD EPYC 7742 CPUs, with each node boasYng 64 cores and 128 threads.The results demonstrate a consistent computaYonal Yme savings of over 30% across all SNR cases.We should note that the computaYonal performance of the modified approach also influences the compression of the decomposed image and enhances the management of memory allocaYon.Although, we demonstrated performance improvement using the modified quantum-inspired approach even on classical compuYng units such as CPU supercomputers, but the full extent of the computaYonal performance could be prominent when applied on quantum compuYng devices.

Remarks
In this paper, we compared an imaging system to an amorphous structure in condensed ma(er physics and used the idea of localizaYon directly to deal with the noisy image.We proposed an algorithm as a complementary revision of previously developed quantum inspired approach for denoising that uYlizes the concept of the localizaYon for filtering process.The modified approach has nearly idenYcal performance to compared with the original method while it compresses the decomposed imaging modes by over 70 %.
One important advantage of the proposed approach is the independence to the set of hyperparameters which were involved in the filtering process and Planck constant value.The Lorentzian funcYon proves to be a robust fit for describing the spectral lines of distribuYon plots and is effecYve to determine the separaYon lines in parYcipaYon raYo vs eigenvalue plots.In regard to Planck constant definiYon, the goal was to select the smallest value to capture all quanYzed image elements.Based on our observaYons, our definiYon provides a sufficiently small value where the localized modes are idenYfiable and the denoising process performs with a good quality.We should note that even smaller value Planck constant results in high localizaYon even for mid-range eigenvalue modes and considerably lower quality denoised images.Despite this trade-off, we maintain confidence in the suitability of our chosen Planck constant definiYon for our imaging system.We note that in previous work 20 a smoothing process on the image was used in order to compute the adapYve basis.This process was necessary when Anderson localizaYon due to high noise levels leads to all wave funcYons being localized on a small part of the system and will be difficult to couple with the present approach.This smoothing process was especially needed for one-dimensional signals, even with relaYvely moderate levels of noise.For two-dimensional systems like images, this regime is present but only at very high noise values, and the present procedure can be safely used up to quite high values of noise.
The modified approach is a standalone algorithm with minimal manual handling of the variables.As menYoned in the recent literature 34,35 , achieving quantum level performance using convenYonal classical algorithms such as machine learning may not be tenable on future quantum computers.This exemplifies the importance of this work where the quantum inspired approach solely relies on quantum physics basics with minimal appeal to convenYonal classical algorithms such as regularizaYon and machine learning.AddiYonally, the concept of localizaYon is not limited to the parYcipaYon raYo characterizaYon, as discussed in previous works 25,27 , there are number of other tools to characterize the amorphous model that might be relevant and applicable to the imaging systems.We purposefully limited the current algorithm to the use of parYcipaYon raYo due to simple required modificaYons, and low computaYonal cost.We should further note that the current modificaYons and depicYon of imaging system as amorphous model is adaptable and can be combined with other previous developments of quantum inspired approaches [36][37][38][39] .

Conclusions
In conclusion, this paper presents a novel quantum-inspired approach for denoising images, leveraging the concept of localizaYon within an amorphous model framework.Building upon previous work, the proposed algorithm demonstrates comparable performance to exisYng methods while significantly reducing the computaYonal complexity and eliminaYng the need for manual intervenYon in parameter tuning.Furthermore, the adopYon of quantum principles aligns with the growing interest in quantum compuYng and offers promising avenues for addressing noise-related challenges across diverse imaging modaliYes both on system level and component level.The former refers to reconstructed image resoluYon in nuclear medicine modaliYes especially in count starved geometries such as limited angle tomography using Yme of flight Positron Emission Tomography 10,11,40,41 , organ-specific Single Photon Emission Computed Tomography 14,42,43 .On detector level implementaYon, quantum-inspired denoising can play a significant role in nuclear imaging detectors during event posiYoning esYmaYon 44 , and detector projecYon processing for novel unconvenYonal detector designs [45][46][47][48] .By embracing quantum principles without heavy reliance on addiYonal classical algorithms, our standalone approach presents a promising path for using quantum-inspired tools in image processing and could lead to new quantum compuYng algorithms in the line of recent proposals 38 .

Figure. 3 .
Figure. 3. Benchmark syntheYc image, and noisy images with Poisson noise and SNR of 2, 5, and 15 are shown in the top.The bo(om shows the corresponding the normalized parYcipaYon raYo [normalized by the total number of pixels] plots of the clean and noisy images.

Figure 4 .
Figure 4. Sample eigenvectors of the decomposed image signal for SNR=2 at (a) low, (b) mid, and (c) high eigenvalues.Each row shows three cross-secYons of an eigenvector.

Fig. 5 .
Fig. 5. Outset plot shows the distribuYon of the normalized parYcipaYon raYo of the syntheYc image with the fi(ed Lorentzian funcYon at the peak of the distribuYon.The inset plot shows the zoom-in view of the selected regions that was used to extract the localized modes at low and high eigenvalue ranges.

Figure 6 .
Figure 6.Comparison between the denoised results of the (i) modified and (ii) original approaches at SNR's of (a) 2 (b) 5 (c) 15.Corresponding normalized parYcipaYon raYo plots with the shaded regions which indicate the filtered regions in modified approach and corresponding histogram of the normalized parYcipaYon raYo are shown in (iii) and (iv) respecYvely.

Figure 7 .
Figure 7. Normalized parYcipaYon raYo plots of the syntheYc image with SNR=2 with augmented Planck constant ℏ 6 for (a)  =1 (b)  = 2, and (c)  =4.The results show that the localizaYon or the disYncYon between different regimes is largely diminished as Planck constant increases.In physics sense, a large Planck constant implies that the Planck length can become larger than the corresponding localizaYon length.Our results show that the choice of Planck constant plays an important role in the idenYficaYon of the different modes.

Figure 9 .
Figure 9. Normalized parYcipaYon raYo plots of noisy Lena images.Shaded regions approximately show the corresponding low and high eigenvalue parts that are truncated in modified approach.

Figure 10 .
Figure 10.ComparaYve analysis of denoising methods: quantum inspired approach with all modes vs. modified Approach with selected high localizaYon modes for CT Images.The figure depicts (a) the original CT image, (b) normalized parYcipaYon

Table 1 .
shows the comparison of SSIM and PSNR between two methods.Comparison SSIM and PSNR for the syntheYc Image

Table 2 .
Comparison SSIM and PSNR for the CT Image

Table 3 .
ComputaYonal Ymes for the CT Image